Finite Group Databasebeta

← #30649193

Subgroups — #30649193

Order 16256.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C212yesmaximal, complemented
2C2×#16861236C212yesmaximal, complemented
3#22550427C211yesmaximal, characteristic, complemented
4#124×D127C211yesmaximal, characteristic, complemented
5#22550268C211yesmaximal, characteristic, complemented
6#22550266C211yesmaximal, characteristic, complemented
7#22550265C211yesmaximal, characteristic, complemented
8#22550261C212yesmaximal, complemented
9#22550257C212yesmaximal, complemented
10#22549828C211yesmaximal, characteristic, complemented
11#497×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×Q8×D127C2211yescharacteristic
14C2×D4×D127C2211yescharacteristic
15#16861241—28no—
16#16861236—28no—
17C2×#14523238C2212yescomplemented
18C2×#14523237C2212yescomplemented
19C2×#14523235C2212yes—
20C2×#14523235C2212yescomplemented
21C2×D1016C2212yes—
22C2×#14523233C2212yes—
23C2×#14523231C2212yes—
24#16861206C2211yescharacteristic, complemented
25#16861198C2211yescharacteristic
26#16861198C2212yescomplemented
27#16861197C2211yescharacteristic
28#16861196C2211yescharacteristic
29#16861195—22no—
30#16861184—22no—
31#16861171—22no—
32C22⋊C4×D127—22no—
33C42×D127C2211yescharacteristic
34#16861120C2211yescharacteristic, complemented
35C8×Dic127C2212yescomplemented
36C2×Q16×C127C2212yescomplemented
37C2×SD16×C127C2212yes—
38C4⋊Q8×C127C2211yescharacteristic, complemented
39#124×C127C2211yescharacteristic
40#97×C127C2211yescharacteristic, complemented
41C23×D127—21no—
42#14523249—24no—
43#14523249—24no—
44C2×Dic254—21no—
45C2×Dic254C2311yescharacteristic
46Q8×D127—24no—
47C2×D508—22no—
48C2×D508—22no—
49C2×D508C2311yescharacteristic
50C2×D508C2312yes—
51C2×C22⋊D127—21no—
52D4×D127—24no—
53C2×C4×D127—21no—
54C2×C4×D127—22no—
55C2×C4×D127D412yes—
56C2×C4×D127C2311yescharacteristic
57#14523238—24no—
58#14523237—24no—
59#14523235—24no—
60#14523235—24no—
61D1016—24no—
62#14523233—24no—
63#14523231D418yescomplemented
64C2×C127⋊C8C2312yes—
65C22⋊Dic127—22no—
66(C2×D127)⋊C4—24no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69C4×Dic127D412yes—
70C4×Dic127C2311yescharacteristic
71C2×Q8×C127C2311yescharacteristic
72C2×Q8×C127C2312yes—
73C2×D4×C127C2311yescharacteristic
74Q16×C127—24no—
75SD16×C127—24no—
76C2×C8×C127C2312yes—
77C4⋊C4×C127—22no—
78C22⋊C4×C127—22no—
79C42×C127C2311yescharacteristic
80C22×D127—21no—
81C22×D127—22no—
82C22×D127—44no—
83C22×D127—42no—
84C22×D127C2×D411yescharacteristic
85Dic254—22no—
86Dic254—42no—
87D508—22no—
88D508—22no—
89D508—22no—
90D508—44no—
91D508—44no—
92C22⋊D127—42no—
93C2×Dic127—21no—
94C2×Dic127C2×D412yes—
95C2×Dic127C2×D411yescharacteristic
96C4×D127—42no—
97C4×D127—42no—
98C4×D127—44no—
99C4×D127C2×D414yes—
100C127⋊C8C2×D414yes—
101C23×C127—21no—
102Q8×C127—22no—
103Q8×C127—22no—
104Q8×C127—22no—
105D4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2411yescharacteristic
110C8×C127C2×D414yes—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127C22×D411yescharacteristic
114C2×D127—22no—
115C2×D127—42no—
116C2×D127—41no—
117C2×D127—44no—
118C2×D127—42no—
119C2×D127—84no—
120C2×D127#12712yes—
121C4×C127—22no—
122C4×C127—42no—
123C4×C127—41no—
124C4×C127C22×D412yes—
125Dic127—22no—
126Dic127—41no—
127Dic127#12712yes—
128C2×C127—81no—
129C2×C127C2×#12711yescharacteristic
130C2×C127#57512yes—
131D127—22no—
132D127—82no—
133D127—81no—
134#3153—1271nomaximal, Sylow
135C127#315311yescharacteristic, complemented, Sylow
136C2×#137—1272no—
137C2×AGL(1,ℤ/8)—1272no—
138#532—1271no—
139C2×#124—1271no—
140#497—1271no—
141#497D12711yescharacteristic, complemented
142#496—1271no—
143#496—1271no—
144#495—1272no—
145#495—1272no—
146C4×M16—1271no—
147C2×#41—1272no—
148C22×Q8—1271no—
149C22×D4—1271no—
150#137—2548no—
151AGL(1,ℤ/8)—2548no—
152C2×Q16—1272no—
153C2×Q16C2×D12712yescomplemented
154C2×SD16—1272no—
155C2×SD16—1272no—
156C2×SD16—1272no—
157C2×SD16C2×D12712yes—
158C2×D8—1272no—
159C2×D8—1272no—
160C2×M16—1272no—
161C4⋊Q8C2×D12711yescharacteristic, complemented
162#127—1271no—
163#124—1271no—
164#124—1272no—
165#124—1271no—
166#124—1271no—
167#124—2542no—
168#124C2×D12711yescharacteristic
169#121—2542no—
170C4×D4—2542no—
171C2×C22⋊C4—2542no—
172C2×C42—1271no—
173#97—1271no—
174#97C2×D12711yescharacteristic, complemented
175C4×C8—1272no—
176C24—2541no—
177#41—2544no—
178#41—2544no—
179C2×Q8—1271no—
180C2×Q8—2541no—
181C2×Q8—2544no—
182C2×Q8C22×D12711yescharacteristic
183C2×Q8C22×D12712yes—
184C2×D4—1271no—
185C2×D4—1272no—
186C2×D4—2542no—
187C2×D4—2544no—
188C2×D4—2541no—
189C2×D4—2542no—
190C2×D4C22×D12711yescharacteristic
191C22×C4—1271no—
192C22×C4—1272no—
193C22×C4—2542no—
194C22×C4—2541no—
195Q16—24no—
196Q16—2544no—
197SD16—24no—
198SD16—2544no—
199SD16—2544no—
200SD16—2544no—
201D8—2544no—
202D8—2544no—
203M16—1278no—
204C2×C8—1272no—
205C2×C8C22×D12712yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—2542no—
209C22⋊C4—2544no—
210C22⋊C4—2542no—
211C22⋊C4—2542no—
212C42—1272no—
213C42—1271no—
214C42C22×D12711yescharacteristic
215C23—21no—
216C23—1271no—
217C23—5082no—
218C23—5084no—
219C23—2541no—
220C23—2542no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—5082no—
225Q8—2542no—
226D4—22no—
227D4—5084no—
228D4—5084no—
229D4—5082no—
230D4—2542no—
231D4—2542no—
232D4—2542no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—1271no—
236C2×C4—1274no—
237C2×C4—1272no—
238C2×C4—5082no—
239C2×C4—5084no—
240C2×C4—5082no—
241C2×C4—2541no—
242C2×C4D4×D12712yes—
243C2×C4C23×D12711yescharacteristic
244C8—1274no—
245C8D4×D12714yes—
246C22—41no—
247C22—42no—
248C22—1272no—
249C22—10164no—
250C22—5082no—
251C22—5081no—
252C22—5084no—
253C22—5082no—
254C22—2542no—
255C22C2×D4×D12711yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—1272no—
260C4—5081no—
261C4—2542no—
262C4C2×D4×D12712yes—
263C2—81no—
264C2—10161no—
265C2—10162no—
266C2—2542no—
267C2#127×D12711yescharacteristic
268C2#2255067512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#497×C127C2
socleC22×C127C22×D4
hypercenter#497D127
hypocenterC127#3153
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C2×#16861236C2semidirect
3. #22550427C2semidirect
4. #124×D127C2semidirect
5. #22550268C2semidirect
6. #22550266C2semidirect
7. #22550265C2semidirect
8. #22550261C2semidirect
9. #22550257C2semidirect
10. #22549828C2semidirect
11. #497×C127C2semidirect
17. C2×#14523238C22semidirect
18. C2×#14523237C22semidirect
20. C2×#14523235C22semidirect
24. #16861206C22semidirect
26. #16861198C22semidirect
34. #16861120C22semidirect
35. C8×Dic127C22semidirect
36. C2×Q16×C127C22semidirect
38. C4⋊Q8×C127C22semidirect
40. #97×C127C22semidirect
63. #14523231D4semidirect
135. C127#3153semidirect
141. #497D127semidirect
153. C2×Q16C2×D127semidirect
161. C4⋊Q8C2×D127semidirect
174. #97C2×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168612412
C2×#168612362
#225504272
#124×D1272
#225502682
#225502662
#225502652
#225502612
#225502572
#225498282
#497×C1272
#3153127